ISSN 1364-0380 (on line) 1465-3060 (printed) 67 Geometry & Topology G T GG T T T Volume 3 (1999) 67–101 G T G T G T G T Published: 28 May 1999 T G T G T G T G T G G G G T T Embeddings from the point of view of immersion theory : Part I Michael Weiss Department of Mathematics, University of Aberdeen Aberdeen, AB24 3UE, UK Email: [email protected] Abstract Let M and N be smooth manifolds without boundary. Immersion theory suggests that an understanding of the space of smooth embeddings emb(M, N) should come from an analysis of the cofunctor V 7→ emb(V, N) from the poset O of open subsets of M to spaces. We therefore abstract some of the properties of this cofunctor, and develop a suitable calculus of such cofunctors, Goodwillie style, with Taylor series and so on. The terms of the Taylor series for the cofunctor V 7→ emb(V, N) are explicitly determined. In a sequel to this paper, we introduce the concept of an analytic cofunctor from O to spaces, and show that the Taylor series of an analytic cofunctor F converges to F . Deep excision theorems due to Goodwillie and Goodwillie–Klein imply that the cofunctor V 7→ emb(V, N) is analytic when dim(N) − dim(M) ≥ 3. AMS Classification numbers Primary: 57R40 Secondary: 57R42 Keywords: Embedding, immersion, calculus of functors Proposed: Ralph Cohen Received: 10 May 1998 Seconded: Haynes Miller, Gunnar Carlsson Revised: 5 May 1999 c Geometry & Topology Publications 68 Michael Weiss 0 Introduction Recently Goodwillie [9], [10], [11] and Goodwillie–Klein [12] proved higher ex- cision theorems of Blakers–Massey type for spaces of smooth embeddings. In conjunction with a calculus framework, these lead to a calculation of such spaces when the codimension is at least 3. Here the goal is to set up the calculus frame- work. This is very similar to Goodwillie’s calculus of homotopy functors [6], [7], [8], but it is not a special case. Much of it has been known to Goodwillie for a long time. For some history and a slow introduction, see [23]. If a reckless introduction is required, read on—but be prepared for Grothendieck topologies [18] and homotopy limits [1], [23, section 1]. Let M and N be smooth manifolds without boundary. Write imm(M, N) for the space of smooth immersions from M to N . Let O be the poset of open subsets of M , ordered by inclusion. One of the basic ideas of immersion theory since Gromov [14], [16], [19] is that imm(M, N) should be regarded as just one value of the cofunctor V 7→ imm(V, N) from O to spaces. Here O is treated as a category, with exactly one morphism V → W if V ⊂ W , and no morphism if V 6⊂ W ; a cofunctor is a contravariant functor. The poset or category O has a Grothendieck topology [18, III.2.2] which we denote by J1 . Namely, a family of morphisms {Vi → W | i ∈ S} qualifies as a covering in J1 if every point of W is contained in some Vi . More generally, for each k > 0 there is a Grothendieck topology Jk on O in which a family of morphisms {Vi → W | i ∈ S} qualifies as a covering if every finite subset of W with at most k elements is contained in some Vi . We will say that a cofunctor F from O to spaces is a homotopy sheaf with respect to the Grothendieck topology Jk if for any covering {Vi → W | i ∈ S} in Jk the canonical map F (W ) −→ holim F (∩i∈RVi) ∅6=R⊂S is a homotopy equivalence. Here R runs through the finite nonempty subsets of S . In view of the homotopy invariance properties of homotopy inverse limits, the condition means that the values of F on large open sets are sufficiently determined for the homotopy theorist by the behavior of F on certain small open sets; however, it depends on k how much smallness we can afford.— The main theorem of immersion theory is that the cofunctor V 7→ imm(V, N) from O to spaces is a homotopy sheaf with respect to J1 , provided dim(N) is greater than dim(M) or dim(M) = dim(N) and M has no compact component. In this form, the theorem may not be very recognizable. It can be decoded as follows. Let Z be the space of all triples (x,y,f) where x ∈ M , y ∈ N and f: TxM → TyN is a linear monomorphism. Let p: Z → M be the projection Geometry & Topology, Volume 3 (1999) Embeddings from immersion theory : I 69 to the first coordinate. For V ∈ O we denote by Γ(p ; V ) the space of partial sections of p defined over V . It is not hard to see that V 7→ Γ(p ; V ) is a homotopy sheaf with respect to J1 . (Briefly: if {Vi → W } is a covering in J1 , then the canonical map q: hocolimR ∩i∈RVi → W is a homotopy equivalence ∗ ∼ according to [24], so that Γ(p ; W ) ≃ Γ(q p) = holimR Γ(p ; ∩i∈RVi) . ) There is an obvious inclusion ( imm(V, N) ֒→ Γ(p ; V (∗) which we regard as a natural transformation between cofunctors in the variable V . We want to show that (∗) is a homotopy equivalence for every V , in partic- ular for V = M ; this is the decoded version of the main theorem of immersion theory, as stated in Haefliger–Poenaru [15] for example (in the PL setting). By inspection, (∗) is indeed a homotopy equivalence when V is diffeomorphic to Rm . An arbitrary V has a smooth triangulation and can then be covered by the open stars Vi of the triangulation. Since (∗) is a homotopy equivalence for the Vi and their finite intersections, it is a homotopy equivalence for V by the homotopy sheaf property. Let us now take a look at the space of smooth embeddings emb(M, N) from the same point of view. As before, we think of emb(M, N) as just one value of the cofunctor V 7→ emb(V, N) from O to spaces. The cofunctor is clearly not a homotopy sheaf with respect to the Grothendieck topology J1 , except in some very trivial cases. For if it were, the inclusion (∗∗) emb(V, N) −→⊂ imm(V, N) would have to be a homotopy equivalence for every V ∈ O, since it is clearly a homotopy equivalence when V is diffeomorphic to Rm . In fact it is quite appro- priate to think of the cofunctor V 7→ imm(V, N) as the homotopy sheafification of V 7→ emb(V, N), again with respect to J1 . The natural transformation (∗∗) has a suitable universal property which justifies the terminology. Clearly now is the time to try out the smaller Grothendieck topologies Jk on O. For each k > 0 the cofunctor V 7→ emb(V, N) has a homotopy sheafification with respect to Jk . Denote this by V 7→ Tk emb(V, N). Thus V 7→ Tk emb(V, N) is a homotopy sheaf on O with respect to Jk and there is a natural transformation (∗∗∗) emb(V, N) −→ Tk emb(V, N) which should be regarded as the best approximation of V 7→ emb(V, N) by a cofunctor which is a homotopy sheaf with respect to Jk . I do not know of any convincing geometric interpretations of Tk emb(V, N) except of course in the case k = 1, which we have already discussed. As Goodwillie explained to me, Geometry & Topology, Volume 3 (1999) 70 Michael Weiss his excision theorem for diffeomorphisms [9], [10], [11] and improvements due to Goodwillie–Klein [12] imply that (∗∗∗) is (k(n − m − 2) + 1 − m)–connected where m = dim(M) and n = dim(N). In particular, if the codimension n − m is greater than 2, then the connectivity of (∗∗∗) tends to infinity with k. The suggested interpretation of this result is that, if n−m> 2, then V 7→ emb(V, N) behaves more and more like a homotopy sheaf on O, with respect to Jk , as k tends to infinity. Suppose now that M ⊂ N , so that emb(V, N) is a based space for each open V ⊂ M . Then the following general method for calculating or partially calcu- lating emb(M, N) is second to none. Try to determine the cofunctors V 7→ homotopy fiber of [Tk emb(V, N) → Tk−1 emb(V, N)] for the first few k> 0. These cofunctors admit a surprisingly simple description in terms of configuration spaces; see Theorem 8.5, and [23]. Try to determine the extensions (this tends to be very hard) and finally specialize, letting V = M . This program is already outlined in Goodwillie’s expanded thesis [9, section In- tro.C] for spaces of concordance embeddings (a special case of a relative case), with a pessimistic note added in revision: “ . it might never be [written up] ...”. It is also carried out to some extent in a simple case in [23]. More details on the same case can be found in Goodwillie–Weiss [13]. Organization Part I (this paper) is about the series of cofunctors V 7→ Tk emb(V, N), called the Taylor series of the cofunctor V 7→ emb(V, N). It is also about Taylor series of other cofunctors of a similar type, but it does not address convergence questions. These will be the subject of Part II ([13], joint work with Goodwillie). Convention Since homotopy limits are so ubiquitous in this paper, we need a “convenient” category of topological spaces with good homotopy limits.